Classifying compact 4-manifolds via generalized regular genus and G-degree

Abstract

$(d+1)$-colored graphs, i.e., edge-colored graphs that are $(d+1)$-regular, have already been proved to be a useful representation tool for compact PL $d$-manifolds, thus extending the theory (known as crystallization theory) originally developed for the closed case. In this context, combinatorially defined PL invariants play a relevant role. The present paper focuses in particular on generalized regular genus and G-degree: the first one extending to higher dimension the classical notion of Heegaard genus for 3-manifolds, the second one arising, within theoretical physics, from the theory of random tensors as an approach to quantum gravity in dimension greater than two. We establish several general results concerning the two invariants, in relation with invariants of the boundary and with the rank of the fundamental group, as well as their behaviour with respect to connected sums. We also compute both generalized regular genus and G-degree for interesting classes of compact $d$-manifolds, such as handlebodies, products of closed manifolds by the interval and ${\mathbb{D}}^2$-bundles over ${\mathbb{S}}^2.$ The main results of the paper concern dimension 4, where we obtain the classification of all compact PL manifolds with generalized regular genus at most one, and of all compact PL manifolds with G-degree at most 18; moreover, in case of empty or connected boundary, the classifications are extended to generalized regular genus two and to G-degree 24.